Little q-Legendre Polynomials and Irrationality of Certain Lambert Series

Abstract

Certain q-analogs hp(1) of the harmonic series, with p = 1/q an integer greater than one, were shown to be irrational by Erdős (J. Indiana Math. Soc.12, 1948, 63–66). In 1991–1992 Peter Borwein (J. Number Theory37, 1991, 253–259; Proc. Cambridge Philos. Soc.112, 1992, 141–146) used Padé approximation and complex analysis to prove the irrationality of these q-harmonic series and of q-analogs lnp(2) of the natural logarithm of 2. Recently Amdeberhan and Zeilberger (Adv. Appl. Math.20, 1998, 275–283) used the qEKHAD symbolic package to find q-WZ pairs that provide a proof of irrationality similar to Apéry's proof of irrationality of ζ(2) and ζ(3). They also obtain an upper bound for the measure of irrationality, but better upper bounds were earlier given by Bundschuh and Väänänen (Compositio Math.91, 1994, 175–199) and recently also by Matala-aho and Väänänen (Bull. Australian Math. Soc.58, 1998, 15–31) (for lnp(2)). In this paper we show how one can obtain rational approximants for hp(1) and lnp(2) (and many other similar quantities) by Padé approximation using little q-Legendre polynomials and we show that properties of these orthogonal polynomials indeed prove the irrationality, with an upper bound of the measure of irrationality which is as sharp as the upper bound given by Bundschuh and Väänänen for hp(1) and a better upper bound as the one given by Matala-aho and Väänänen for lnp(2).